Fuzzy Extension Principle and Fuzzy Arithmetic and Fuzzy Arithmetic

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Fuzzy Extension Principle and Fuzzy Arithmetic

and Fuzzy Arithmetic

Lecture 06 Lecture 06

Extension Principle for

Crisp Sets

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Crisp Mapping

Extension Principle for Crisp Sets

A mapping can also be expressed by a realtion R on the Cartesian space XxY with the characteristic function:

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Extension Principle for Crisp Sets

Let A be a crisp set defined on X. The mapping y=f(x) will result in a set B defined on Y such that

and the characteristic function of B will be

Here, B is another crisp set.

Extension Principle for Crisp Sets

Example: Let X={ -2, -1, 0, 1, 2} and A={ 0, 1} defined on X.

y= |4x|+2 mapping is applied to A, find B.

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Extension Principle for Crisp Sets

Method #1:

Directly appliying the formula:

Extension Principle for

Crisp Sets

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Extension Principle for Crisp Sets

Method #2:

Use relation matrix

Then, B = A o R

Fuzzy Mapping

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Extension Principle for Fuzzy Sets

~

~ ~

~

:

( ) :

: .

.

A fuzzy set defined on X

y f x functional transform or mapping B image of A on X under f

B is a fuzzy set having universe of discourse Y

=

( )

~ ~

B

=

f A

~

Fuzzy Extension Principle

General definition: Suppose f is a mapping from an n-dimentional Cartesian product space X₁×X₂×… × X_nto a one dimentional universe Y such that and suppose A₁,A₂,…,A_nare n fuzzy sets in x₁,x₂, .., x_nrespectively. Then, the image of

A₁,A₂,…,A_nunder f is given as:

~ ~ ~

{

^~¹^{( )}¹ ^~²^{( )}² ^~^{( )}

}

( ) max min( , ,..., ,)

n n

B

y

A x A x A x

μ = μ μ μ

Zadeh’s extension principle

1 2

( , ,..., )_n f x x x =y

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Fuzzy Extension Principle

Example:

Fuzzy Extension Principle

Example:

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Example: cont.

Fuzzy Extension Principle

Definition: A fuzzy vector is a vector containing fuzzy membership values.

can be determined directly fromy using vector form:g where is an n×m fuzzy relation matrix

Composition: max-min

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Fuzzy Extension Principle

Definition: If the input is a sigle element(a fuzzy singleton), the image of this singleton will be fuzzy and this case is termed as fuzzy transform.

Fuzzy Extension Principle

Fuzzy transform of the singleton is given by the row of the fuzzy relation R.

Definition: For a function f that performs a one-to-one mapping (i.e.,maps one element in universe U to one element in universe V),

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Mapping of more than one input variable:

Suppose and inputs are mapped to V through , If the mapping is one-to-one, the same membership grades results but if the mapping is not one-to-one , maximum membership grades maping to the same output variable is accepted.

Fuzzy Arithmetic and Fuzzy Numbers:

Let and be two fuzzy numbers with defined on X and defined on Y, and let the symbol * denote a general arithmetic operation.

I~

*

Fuzzy Extension Principle

An arithmetic operation between these two fuzzy numbers, denoted is a mapping to another universe, say Z, and accomplished by using the extention principle:

Example:

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Fuzzy Extension Principle

Let’s map the product of and to a fuzzy number Extension principle:

Fuzzy Extension Principle

0

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Fuzzy Extension Principle

Example:

Fuzzy Extension Principle

Example: